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Questions and Answers of Statistics

Find each of these values? a) (992 mod 32)3 mod 15 b) (34 mod 17)2 mod 11 c) (193 mod 23)2 mod 31 d) (893 mod 79)4 mod 26
Show that if n | m, where n and m are integers greater than 1, and if a ≡ b (mod m), where a and b are integers, then a ≡ b (mod n)?
Find counterexamples to each of these statements about congruences.a) If ac ≡ bc (mod m), where a, b, c, and m are integers with m ≥ 2, then a ≡ b (mod m).b) If a ≡ b (mod m) and c ≡ d (mod
Use Exercise 38 to show that if m is a positive integer of the form 4k + 3 for some nonnegative integer k, then m is not the sum of the squares of two integers?
Show that if a, b, k, and m are integers such that k ≥ 1, m ≥ 2, and a ≡ b (mod m), then ak ≡ bk(mod m)?
Show that Zm with multiplication modulo m, where m ≥ 2 is an integer, satisfies the closure, associative, and commutatively properties, and 1 is a multiplicative identity?
Write out the addition and multiplication tables for Z5 (where by addition and multiplication we mean +5 and ·5)?
Determine whether each of the functions f (a) = a div d and g(a) = a mod d, where d is a fixed positive integer, from the set of integers to the set of integers, is one-to-one, and determine whether
Show that if a | b and b | a, where a and b are integers, then a = b or a = - b?
Show that if a, b, and c are integers, where a ( 0 and c ( 0, such that ac | bc, then a | b?
What are the quotient and remainder when a) 19 is divided by 7? b) −111 is divided by 11? c) 789 is divided by 23? d) 1001 is divided by 13? e) 0 is divided by 19? f) 3 is divided by 5? g) −1 is
Convert the decimal expansion of each of these integers to a binary expansion? a) 231 b) 4532 c) 97644
Show that the hexadecimal expansion of a positive integer can be obtained from its binary expansion by grouping together blocks of four binary digits, adding initial zeros if necessary, and
Show that the octal expansion of a positive integer can be obtained from its binary expansion by grouping together blocks of three binary digits, adding initial zeros if necessary, and translating
Convert (7345321)8 to its binary expansion and (10 1011 1011)2 to its octal expansion?
Give a procedure for converting from the octal expansion of an integer to its hexadecimal expansion using binary notation as an intermediate step?
Find the sum and the product of each of these pairs of numbers. Express your answers as a binary expansion? a) (100 0111)2, (111 0111)2 b) (1110 1111)2, (1011 1101)2 c) (10 1010 1010)2, (1 1111
Find the sum and product of each of these pairs of numbers? Express your answers as an octal expansion. a) (763)8, (147)8 b) (6001)8, (272)8 c) (1111)8, (777)8 d) (54321)8, (3456)8
Use Algorithm 5 to find 7644 mod 645?
Use Algorithm 5 to find 32003 mod 99?
Show that every positive integer can be represented uniquely as the sum of distinct powers of 2?
Convert the binary expansion of each of these integers to a decimal expansion? a) (1 1111)2 b) (10 0000 0001)2 c) (1 0101 0101)2 d) (110 1001 0001 0000)2
Show that a positive integer is divisible by 3 if and only if the sum of its decimal digits is divisible by 3?
Show that a positive integer is divisible by 3 if and only if the difference of the sum of its binary digits in evennumbered positions and the sum of its binary digits in odd-numbered positions is
What integer does each of the following one's complement representations of length five represent? a) 11001 b) 01101 c) 10001 d) 11111
How is the one's complement representation of the sum of two integers obtained from the one's complement representations of these integers?
Show that the integer m with one's complement representation (an−1an−2 . . . a1a0) can be found using the equation m = −an−1(2n−1 − 1) + an−22n−2 + · · · + a1 · 2 + a0?
Exercise 35 if each expansion is a two's complement expansion of length five.
Exercise 38 for two's complement expansions?
Convert the octal expansion of each of these integers to a binary expansion? a) (572)8 b) (1604)8 c) (423)8 d) (2417)8
Convert the hexadecimal expansion of each of these integers to a binary expansion? a) (80E)16 b) (135AB)16 c) (ABBA)16 d) (DEFACED)16?
Convert (ABCDEF)16 from its hexadecimal expansion to its binary expansion?
Determine whether each of these integers is prime. a) 21 b) 29 c) 71 d) 97 e) 111 f) 143
Show that log2 3 is an irrational number. Recall that an irrational number is a real number x that cannot be written as the ratio of two integers.
Prove or disprove that there are three consecutive odd positive integers that are primes, that is, odd primes of the form p, p + 2, and p + 4.
Which positive integers less than 30 are relatively prime to 30?
Determine whether the integers in each of these sets are pairwise relatively prime. a) 11, 15, 19 b) 14, 15, 21 c) 12, 17, 31, 37 d) 7, 8, 9, 11
Show that if 2n − 1 is prime, then n is prime.
Find these values of the Euler φ-function. a) φ(4). b) φ(10). c) φ(13).
What is the value of φ(pk) when p is prime and k is a positive integer?
What are the greatest common divisors of these pairs of integers? a) 37 · 53 · 73, 211 · 35 · 59 b) 11 · 13 · 17, 29 · 37 · 55 · 73 c) 2331, 2317 d) 41 · 43 · 53, 41 · 43 · 53 e) 313 ·
What is the least common multiple of each pair in Exercise 25?
Find gcd(92928, 123552) and lcm(92928, 123552), and verify that gcd(92928, 123552) · lcm(92928, 123552) = 92928 · 123552.
Find the prime factorization of each of these integers. a) 88 b) 126 c) 729 d) 1001 e) 1111 f) 909,090
Show that if a and b are positive integers, then ab = gcd(a, b) · lcm(a, b).
Use the Euclidean algorithm to find a) gcd(12, 18). b) gcd(111, 201). c) gcd(1001, 1331). d) gcd(12345, 54321). e) gcd(1000, 5040). f) gcd(9888, 6060).
How many divisions are required to find gcd(34, 55) using the Euclidean algorithm?
Use Exercise 36 to show that if a and b are positive integers, then gcd(2a − 1, 2b − 1) = 2gcd(a, b) − 1.
Use the extended Euclidean algorithm to express gcd(26, 91) as a linear combination of 26 and 91. The extended Euclidean algorithm can be used to express gcd(a, b) as a linear combination with
Use the extended Euclidean algorithm to express gcd(144, 89) as a linear combination of 144 and 89. The extended Euclidean algorithm can be used to express gcd(a, b) as a linear combination with
Describe the extended Euclidean algorithm using pseudocode. The extended Euclidean algorithm can be used to express gcd(a, b) as a linear combination with integer coefficients of the integers a and
Can you find a formula or rule for the nth term of a sequence related to the prime numbers or prime factorizations so that the initial terms of the sequence have these values? a) 0, 1, 1, 0, 1, 0, 1,
Prove that the product of any three consecutive integers is divisible by 6.
Prove or disprove that n2 − 79n + 1601 is prime whenever n is a positive integer.
Show that there is a composite integer in every arithmetic progression ak + b, k = 1, 2, . . . where a and b are positive integers.
Adapt the proof in the text that there are infinitely many primes to prove that there are infinitely many primes of the form 4k + 3, where k is a nonnegative integer.
Prove that the set of positive rational numbers is countable by showing that the function K is a one-to-one correspondence between the set of positive rational numbers and the set of positive
Express in pseudocode the trial division algorithm for determining whether an integer is prime.
Show that if am + 1 is composite if a and m are integers greater than 1 and m is odd.
Show that 15 is an inverse of 7 modulo 26.
Solve each of these congruences using the modular inverses found in parts (b), (c), and (d) of Exercise 5. a) 19x ≡ 4 (mod 141) b) 55x ≡ 34 (mod 89) c) 89x ≡ 2 (mod 232)
Find the solutions of the congruence 15x2 + 19x ≡ 5 (mod 11).
Show that if m is an integer greater than 1 and ac ≡ bc (mod m), then a ≡ b (mod m/gcd(c,m)).
Show that if p is prime, the only solutions of x2 ≡ 1 (mod p) are integers x such that x ≡ 1 (mod p) or x ≡ −1 (mod p).
This exercise outlines a proof of Fermat's little theorem. a) Suppose that a is not divisible by the prime p. Show that no two of the integers 1 · a, 2 · a, . . . , (p − 1)a are congruent modulo
Use the construction in the proof of the Chinese remainder theorem to find all solutions to the system of congruences x ≡ 1 (mod 2), x ≡ 2 (mod 3), x ≡ 3 (mod 5), and x ≡ 4 (mod 11).
Solve the system of congruences in Exercise 20 using the method of back substitution.
Write out in pseudocode an algorithm for solving a simultaneous system of linear congruences based on the construction in the proof of the Chinese remainder theorem.
Find all solutions, if any, to the system of congruences x ≡ 7 (mod 9), x ≡ 4 (mod 12), and x ≡ 16 (mod 21).
Let m1, m2, . . . , mn be pairwise relatively prime integers greater than or equal to 2. Show that if a ≡ b (mod mi) for i = 1, 2, . . . , n, then a ≡ b (mod m), where m = m1m2 · · ·mn. (This
By inspection (as discussed prior to Example 1), find an inverse of 4 modulo 9.
Which integers leave a remainder of 1 when divided by 2 and also leave a remainder of 1 when divided by 3?
Use Fermat's little theorem to find 7121 mod 13.
Use Fermat's little theorem to show that if p is prime and p X | a, then ap−2 is an inverse of a modulo p.
a) Show that 2340 ≡ 1 (mod 11) by Fermat's little theorem and noting that 2340 = (210)34. b) Show that 2340 ≡ 1 (mod 31) using the fact that 2340 = (25)68 = 3268. c) Conclude from parts (a) and
Show that if p is an odd prime, then every divisor of the Mersenne number 2p − 1 is of the form 2kp + 1, where k is a nonnegative integer.
Use Exercise 41 to determine whether M11 = 211 − 1 = 2047 and M17 = 217 − 1 = 131,071 are prime.
Show that 2047 is a strong pseudoprime to the base 2 by showing that it passes Miller's test to the base 2, but is composite.
Show that 2821 is a Carmichael number.
a) Use Exercise 48 to show that every integer of the form (6m + 1)(12m + 1)(18m + 1), where m is a positive integer and 6m + 1, 12m + 1, and 18m + 1 are all primes, is a Carmichael number. b) Use
Express each nonnegative integer a less than 15 as a pair (a mod 3, a mod 5).
Solve the system of congruences that arises in Example 8.
Find the discrete logarithms of 5 and 6 to the base 2 modulo 19.
Write out a table of discrete logarithms modulo 17 with respect to the primitive root 3.
Show that if p is an odd prime and a is an integer not divisible by p, then the congruence x2 ≡ a (mod p) has either no solutions or exactly two incongruent solutions modulo p.
Show that if p is an odd prime and a and b are integers with a ‰¡ b (mod p), then
Use Exercise 62 to show that if p is an odd prime and a and b are integers not divisible by p, then
Find all solutions of the congruence x2 ≡ 29 (mod 35).
Describe a brute force algorithm for solving the discrete logarithm problem and find the worst-case and average-case time complexity of this algorithm.
Show that if a and m are relatively prime positive integers, then the inverse of a modulo m is unique modulo m.
Solve the congruence 4x ≡ 5 (mod 9) using the inverse of 4 modulo 9 found in part (a) of Exercise 5.
Which memory locations are assigned by the hashing function h(k) = k mod 97 to the records of insurance company customers with these Social Security numbers? a) 034567981 b) 183211232 c) 220195744 d)
Find the sequence of pseudorandom numbers generated by the power generator with p = 7, d = 3, and seed x0 = 2.
Suppose you received these bit strings over a communications link, where the last bit is a parity check bit. In which string are you sure there is an error? a) 00000111111 b) 10101010101 c)
The first nine digits of the ISBN-10 of the European version of the fifth edition of this book are 0-07-119881. What is the check digit for that book?
Determine whether the check digit of the ISBN-10 for this textbook (the seventh edition of Discrete Mathematics and its Applications) was computed correctly by the publisher.
Determine whether each of these numbers is a valid USPS money order identification number. a) 74051489623 b) 88382013445 c) 56152240784 d) 66606631178
One digit in each of these identification numbers of a postal money order is smudged. Can you recover the smudged digit, indicated by a Q, in each of these numbers? a) 493212Q0688 b) 850Q9103858 c)
Determine which transposition errors are detected by the USPS money order code.
Determine whether each of the strings of 12 digits is a valid UPC code. a) 036000291452 b) 012345678903 c) 782421843014 d) 726412175425

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